Logic transformation and gate placement to avoid routing congestion

ABSTRACT

A novel logic design method for avoiding wiring congestion. According to the novel logic design method, an original gate having multiple inputs coming from different directions and having multiple outputs coming to different directions can be transformed to a logic block that has an input stage and an output stage. The gates of the input stage receive signals from the multiple inputs of the original gate. The gates of the output stage send signals to the multiple outputs of the original gate. Each gate of the input stage is placed in a vicinity of its inputs. Each gate of the output stage is placed in a vicinity of its outputs. The gates of the input and output stages are functionally equivalent to the original gate.

This application is a divisional application claiming priority to Ser.No. 11/153,707, Filed Jun. 14, 2005.

BACKGROUND OF THE INVENTION

1. Technical Field

The present invention relates to logic design, and more particularly, toa logic design method to avoid routing congestion.

2. Related Art

In a conventional digital circuit, a logic block can receive signalsfrom multiple inputs from different directions and send signals tomultiple outputs at different directions. Typically, the logic blocktends to be placed at a central area of the digital circuit surroundedby the inputs and outputs of the logic block. This tends to result inwiring congestion in the central area.

It is always desirable to reduce wiring congestion in logic design.Minimizing wiring congestion improves wirability and reduces theadjacency capacitance impact on timing and noise. As a result, there isa need for a logic design method for reducing wiring congestion in alogic design.

SUMMARY OF THE INVENTION

The present invention provides a logic design method, comprising thesteps of (a) providing a logic block comprising an input stage and anoutput stage; and (b) placing each input gate of the input stage in avicinity of the input gate's respective inputs.

The present invention also provides a logic design method, comprisingthe steps of (a) transforming a gate G into a logic block including aninput stage and an output stage, wherein inputs of the input stage arethe inputs of the gate G, and wherein outputs of the output stage arethe outputs of the gate G; and (b) placing each input gate of the inputstage in a vicinity of the input gate's respective inputs.

The present invention also provides a logic design method, comprisingthe steps of (a) transforming a two-stage logic into the logic blockincluding an input stage and an output stage, wherein the two-stagelogic includes a first stage and a second stage, wherein the first stageincludes T gates, and wherein the second stage includes an output gatethat is configured to receive input signals from all the T gates of thefirst stage; and (b) placing each input gate of the input stage in avicinity of the input gate's respective inputs.

The present invention provides a logic design method for reducing wiringcongestion in a logic design.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1C show logic diagrams used to illustrate a first logic designmethod, in accordance with embodiments of the present invention.

FIG. 1D is a flowchart that illustrates the first logic design method.

FIGS. 2A-2C show logic diagrams used to illustrate a second logic designmethod, in accordance with embodiments of the present invention.

FIG. 2D is a flowchart that illustrates the second logic design method.

FIG. 3 illustrates a flow chart of a logic design method, in accordancewith embodiments of the present invention.

FIG. 4 illustrates a computer system 90 used for simulating the methodsof the present invention.

DETAILED DESCRIPTION OF THE INVENTION

FIGS. 1A-1C show logic diagrams used to illustrate a first logic designmethod, in accordance with embodiments of the present invention. FIG. 1Dis a flowchart 199 that illustrates the first logic design method. Morespecifically, with reference to FIG. 1A, as a first example toillustrate the first logic design method, assume that a gate G receivesinput signals from inputs A1, A2, A3, B1, and B2, and sends outputsignals to outputs O1, O2, O3, O4, and O5. The input signals from theinputs A1, A2, A3, B1, and B2 come to the gate G from differentdirections: North, East, North-West, South, and West, respectively. Theoutput signals from gate G fans-out to outputs O1, O2, O3, O4, and O5 indifferent directions: North-West, North-East, South-East, South-East,and South-West, respectively.

With reference to FIG. 1B, in one embodiment, the first logic designmethod starts with the step of transforming the gate G (FIG. 1A) into alogic block 100 which comprises an input stage 110 and an output stage120. In one embodiment, the inputs of the input stage 110 can be alsothe inputs of the gate G of FIG. 1A (i.e., the inputs A1, A2, A3, B1,and B2), and the outputs of the output stage 120 can be also the outputsof the gate G of FIG. 1A (i.e., the outputs O1, O2, O3, O4, and O5). Inaddition, the input stage 110 is configured to send signals to theoutput stage 120 via connections 125.

In one embodiment, the step of transforming the gate G (FIG. 1A) intothe logic block 100 can comprise the following sub-steps (steps 180-190of FIG. 1D).

In a first sub-step (step 180 of FIG. 1D), in one embodiment, all theinputs of the input stage 110 can be divided into M input groups of atleast two proximate inputs (M is a positive integer). “Proximate” can bedefined in different ways. For instance, in one definition, two or morepoints on a plane can be considered proximate to each other if theradius of the smallest circle within which the points reside is lessthan a pre-specified value. In the first example above, assume thatinputs A1, A2, and A3 are proximate to each other, and that inputs B1and B2 are proximate to each other. As a result, the five inputs A1, A2,A3, B1, and B2 can be divided into M=2 input groups of at least 2proximate inputs: (A1, A2, A3) and (B1, B2).

In a next sub-step (step 182 of FIG. 1D), in one embodiment, all theoutputs of the output stage 120 can be divided into N output groups ofat least two proximate outputs (N is a positive integer). In the firstexample above, assume that outputs O1 and O2 are proximate to eachother, and that outputs O3, O4, and O5 are proximate to each other. As aresult, the five outputs O1, O2, O3, O4, and O5 can be divided into N=2output groups of at least two proximate outputs: (O1, O2) and (O3, O4,O5).

In a next sub-step (step 184 of FIG. 1D), in one embodiment, as a resultof the inputs division above, the input stage 110 can comprise M gates(not shown) corresponding to the M input groups. Similarly, the outputstage 120 can comprise N gates (not shown) corresponding to the N outputgroups (step 186 of FIG. 1D).

In the first example above, with reference to FIGS. 1B and 1C, becausethere are M=2 input groups (A1, A2, A3) and (B1, B2), the input stage110 can comprise M=2 gates Gi1 and Gi2 (FIG. 1C) corresponding to theM=2 input groups (A1, A2, A3) and (B1, B2), respectively. Similarly,because there are N=2 output groups (O1, O2) and (O3, O4, O5), theoutput stage 120 can comprise N=2 gates Go1 and Go2 (FIG. 1C)corresponding to the N=2 output groups (O1, O2) and (O3, O4, O5),respectively.

In a next sub-step (step 188 of FIG. 1D), in one embodiment, connectionsbetween the M gates and the N gates and the inputs and outputs can bemade as follows. Each gate of the M gates of the input stage 110 isconfigured to receive input signals from all the inputs of the gate'scorresponding input group. In the first example above, gate Gi1 isconfigured to receive input signals from the inputs A1, A2, A3.Similarly, gate Gi2 is configured to receive input signals from theinputs B1, B2.

In one embodiment, each gate of the N gates of the output stage 120 isconfigured to receive input signals from all the M gates of the inputstage 110. In the first example above, gate Go1 is configured to receiveinput signals from all the M=2 gates Gi1 and Gi2 of the input stage 110.Similarly, gate Go2 is configured to receive input signals from all theM=2 gates Gi1 and Gi2 of the input stage 110.

Also, in one embodiment, each gate of the N gates of the output stage120 is configured to send output signals to all the outputs of itscorresponding output group. In the first example above, gate Go1 isconfigured to send output signals to the outputs O1 and O2. Similarly,gate Go2 is configured to send output signals to the outputs O3, O4, O4.

In a next sub-step (step 190 of FIG. 1D), in one embodiment, the M gatesand N gates in the input stage 110 and output stage 120, respectivelyare such chosen such that they are functionally equivalent to the gate G(FIG. 1A).

In the first example above, if the gate G (FIG. 1A) is an OR gate, thenthe gates Gi1 and Gi2 and the gates Go1 and Go2 can all be OR gates. Ifthe gate G (FIG. 1A) is an AND gate, then the gates Gi1 and Gi2 and thegates Go1 and Go2 can all be AND gates. Alternatively, if the gate G(FIG. 1A) is an OR gate, then the gates Gi1 and Gi2 of the input stage110 can be NOR gates, and the gates Go1 and Go2 of the output stage 120can be NAND gates. If the gate G (FIG. 1A) is an AND gate, then thegates Gi1 and Gi2 of the input stage 110 can be NAND gates, and thegates Go1 and Go2 of the output stage 120 can be NOR gates.

Next, after the step of transforming the gate G (FIG. 1A) into the logicblock 100 (i.e., steps 180-190 of FIG. 1D), the first logic designmethod continues with the step of placing each gate of the input stage110 in a vicinity of the gate's respective inputs and placing each gateof the output stage 120 in a vicinity of the gate's respective outputs(step 192 of FIG. 1D). “In a vicinity” can be defined in different ways.For instance, a point X can be considered being in a vicinity of pointsX_(i), i=1, . . . , L (L is a positive integer) if ΣD_(i) ², (i=1, . . ., L) does not exceed a pre-specified value, wherein D_(i) is thedistance between X and X_(i).

In the first example above, with reference to FIG. 1C, gate Gi1 of theinput stage 110 can be placed in a vicinity of its respective inputs A1,A2, and A3. Similarly, the gate Gi2 of the input stage 110 can be placedin a vicinity of its respective inputs B1 and B2. Similarly, the gateGo1 of the output stage 120 can be placed in a vicinity of itsrespective outputs O1 and O2. The gate Go2 of the output stage 120 canbe placed in a vicinity of its respective outputs O3, O4, and O5.

As can be seen in FIG. 1C, the central area surrounded by the inputs A1,A2, A3, B1, and B2 and the outputs O1, O2, O3, O4, and O5 is lesscrowded because the replacing gates Gi1, Gi2, Go1, and Go2 are placedclose to their inputs or outputs.

FIGS. 2A-2C show logic diagrams used to illustrate a second logic designmethod, in accordance with embodiments of the present invention. FIG. 2Dis a flowchart 299 that illustrates the second logic design method. Morespecifically, with reference to FIG. 2A, as a second example toillustrate the second logic design method, assume that a two-stage logic200 comprises a first stage 201 and a second stage 202. Assume furtherthat the first stage 201 comprises 2 gates F1 and F2 (in general, thefirst stage 201 can comprise T gates, wherein T is a positive integer),and that the second stage 202 comprises an output gate E that isconfigured to receive input signals from all the 2 gates of the firststage 201.

In the second example above, assume further that the gate F1 has fiveinputs X1, X2, X3, Y1, and Y2, of which inputs X1, X2, and X3 areproximate to each other, and inputs Y1 and Y2 are proximate to eachother. Assume further that the gate F2 has five inputs Z1, Z2, W3, W1,and V, of which inputs Z1 and Z2 are proximate to each other, and inputsW1 and W2 are proximate to each other. Assume further that the outputgate E has five outputs Q1, Q2, Q3, Q4, and Q5 of which outputs Q1, Q2,and Q3 are proximate to each other, and outputs Q4 and Q5 are proximateto each other.

With reference to FIG. 2B, in one embodiment, the second logic designmethod starts with the step of transforming the two-stage logic 200(FIG. 2A) into a logic block 205 which comprises an input stage 210 andan output stage 220. In one embodiment, the step of transforming thetwo-stage logic 200 (FIG. 2A) into the logic block 205 can comprise thefollowing sub-steps (steps 280-284 of FIG. 2D).

In a first sub-step (step 280 of FIG. 2D), in one embodiment, all the Tgates of the first stage 201 (FIG. 2A), i.e., gates F1 and F2 in thesecond example above, are placed in the input stage 210 of the logicblock 205. In other words, the input stage 210 of the logic block 205can comprise the same gates as the first stage 201 (FIG. 2A).

In a next sub-step (step 282 of FIG. 2D), in one embodiment, the outputsof the output gate E (FIG. 2A) can be divided into P output groups of atleast two proximate outputs, wherein P is a positive integer. In thesecond example above, the outputs Q1, Q2, Q3, Q4, and Q5 can be dividedinto P=2 output groups: (Q1, Q2, Q3) and (Q4, Q5) because Q1, Q2, and Q3are proximate to each other, and because Q4 and Q5 are proximate to eachother.

In a next sub-step (step 284 of FIG. 2D), in one embodiment, as a resultof the outputs division above, P gates (not shown) corresponding to theP output groups can be placed in the output stage 220. In oneembodiment, each gate of the P gates is (i) functionally equivalent tothe output gate E (FIG. 2A), (ii) is configured to receive input signalsfrom all the gates of the input stage 210 (i.e., gates F1 and F2), and(iii) is configured to send output signals to all the outputs of thegate's corresponding output group.

In the second example above, P=2 gates E1 and E2 can be placed in theoutput stage 220, wherein each of the gates E1 and E2 is (i)functionally equivalent to the output gate E, (ii) is configured toreceive input signals from both the gates F1 and F2, and (iii) isconfigured to send output signals to all the outputs of the gate'scorresponding output group. More specifically, gate E1 is configured tosend output signals to its outputs Q1, Q2, and Q3. Similarly, gate E2 isconfigured to send output signals to its outputs Q4 and Q5.

Next, in one embodiment, the second logic design method furthercomprises the step of (step 286 of FIG. 2D), placing each gate of theinput stage 210 in a vicinity of the gate's inputs. In the secondexample above, gate F1 is placed in a vicinity of its inputs X1, X2, X3,Y1, and Y2. Similarly, gate F2 is placed in a vicinity of its inputs Z1,Z2, W1, W2, and V.

Next, with reference to FIG. 2C, in one embodiment, the second logicdesign method further comprises the step of (step 288 of FIG. 2D), foreach gate Fi, i=t, . . . , T of the T gates of the input stage 210, (a)dividing inputs of the gate Fi into Qi input groups of at least twoproximate inputs and Ri individual inputs, wherein Qi and Ri arenon-negative integers but are not both equal to zero; (b) replacing thegate Fi by Qi+1 gates, wherein one gate Fi0 of the Qi+1 gates is locatedwhere the gate Fi was, and wherein the other Qi gates of the Qi+1 gatescorrespond to the Qi input groups; and (c) placing each gate of theother Qi gates in a vicinity of the gate's corresponding input group,wherein each gate of the other Qi gates is configured to receive inputsignals from all the inputs of the gate's corresponding input group,wherein the gate Fi0 is configured to receive input signals from all theother Qi gates and from all the Ri individual inputs, and wherein eachgate of the P gates of the output stage is configured to receive inputsignals from all the gates Fi0, i=1, . . . , T.

In the second example above, T=2. For gate F1 (FIG. 2B), its inputs X1,X2, X3, Y1, and Y2 can be divided into Q1=2 input groups: (X1, X2, X3)and (Y1, Y2). It should be noted that R1=0. As a result, gate F1 (FIG.2B) can be replaced by Q1+1=3 gates F10, F11, and F12 (FIG. 2C), whereinF10 is located where the gate F1 was. The gates F11 and F12 correspondto and are placed in a vicinity of the input groups (X1, X2, X3) and(Y1, Y2), respectively. The gate F10 receives signals from gates F11 andF12.

Similarly, for gate F2 (FIG. 2B), its inputs Z1, Z2, W1, W2, and V canbe divided into Q2=2 input groups: (Z1, Z2) and (W1, W2) and R2=1individual input V. As a result, gate F2 (FIG. 2B) can be replaced byQ2+1=3 gates F20, F21, and F22, wherein F20 is located where the gate F2was. The gates F21 and F22 correspond to and are placed in a vicinity ofthe input groups (Z1, Z2) and (W1, W2), respectively. The gate F20receives signals from gates F21 and F22 and also from the individualinput V.

Next, in one embodiment, the functions of the gates in the input stage220 are determined (step 290 of FIG. 2D). In one embodiment, thereplacing gates can be functionally equivalent to the gate they replace.More specifically, for i=1, . . . , T, if the gate Fi is an OR gate,then all the Qi gates and the gate Fi0 can be OR gates; and if the gateFi is an AND gate, then all the Qi gates and the gate Fi0 can be ANDgates.

In the second example above, if gate F1 is an OR gate, then gates F10,F11, and F12 can be OR gates. If gate F1 is an AND gate, then gates F10,F11, and F12 can be AND gates. Similarly, if gate F2 is an OR gate, thengates F20, F21, and F22 can be OR gates. If gate F2 is an AND gate, thengates F20, F21, and F22 can be AND gates.

Alternatively, in one embodiment, for i=1, . . . , T, if the gate Fi isan OR gate, then all the Qi gates can be NOR gates, and the gate Fi0 canbe a NAND gate; and if the gate Fi is an AND gate, then all the Qi gatescan be NAND gates, and the gate Fi0 can be a NOR gate.

In the example above, if gate F1 is an OR gate, then F11 and F12 can beNOR gates, and the gate F10 can be a NAND gate; and if the gate F1 is anAND gate, then gates F11 and F12 can be NAND gates, and the gate F10 canbe a NOR gate. Similarly, if gate F2 is an OR gate, then F21 and F22 canbe NOR gates, and the gate F20 can be a NAND gate; and if the gate F2 isan AND gate, then gates F21 and F22 can be NAND gates, and the gate F20can be a NOR gate.

Next, in one embodiment, the second logic design method furthercomprises the step of placing each gate of the output stage 220 in avicinity of the gate's corresponding outputs (step 292 of FIG. 2D). Inthe second example above, gate E1 is placed in a vicinity of outputs Q1,Q2, and Q3. Similarly, gate E2 is placed in a vicinity of outputs Q4 andQ5.

As can be seen in FIG. 2C, the central area surrounded by the inputs X1,X2, X3, Y1, Y2, Z1, Z2, W1, W2, and V and the outputs Q1, Q2, Q3, Q4,and Q5 is less crowded because the replacing gates are placed close totheir inputs or outputs.

In the embodiments described above, the transformed logic block cancomprise one gate (e.g., gate G of FIG. 1A) or two levels of gates (thetwo-stage logic 200 of FIG. 2A). In general, if an original logic blockhas more than two levels of gates, the original logic block can be firstreduced to two levels of gates using any conventional transformationmethods. Then, the second logic design method of the present inventioncan be applied.

FIG. 3 illustrates a flow chart of a logic design method 300 thatutilizes the present invention. The logic design method 300 starts witha step 310 in which a circuit is described at a high level (i.e.,circuit component diagram). Next, in step 320, the high level circuitdescription is synthesized (i.e., converted) into a list of logic gatesand their interconnections called a netlist. Next, in step 330, thelocations of the logic gates are determined. Next, in step 340, therouting scheme and congestion are analyzed. Next, in step 350, based onthe analysis in step 340, logic blocks with congestion can beidentified. Then, in step 360, the methods of the present inventiondescribed in the embodiments above can be applied to the congested logicblocks.

The methods of the present invention can be applied to selective logicalblocks during physical design. This does not require synthesis placementiteration.

FIG. 4 illustrates a computer system 90 used for simulating the methodsof the present invention. The computer system 90 comprises a processor91, an input device 92 coupled to the processor 91, an output device 93coupled to the processor 91, and memory devices 94 and 95 each coupledto the processor 91. The input device 92 may be, inter alia, a keyboard,a mouse, etc. The output device 93 may be, inter alia, a printer, aplotter, a computer screen, a magnetic tape, a removable hard disk, afloppy disk, etc. The memory devices 94 and 95 may be, inter alia, ahard disk, a floppy disk, a magnetic tape, an optical storage such as acompact disc (CD) or a digital video disc (DVD), a dynamic random accessmemory (DRAM), a read-only memory (ROM), etc. The memory device 95includes a computer code 97. The computer code 97 includes an algorithmfor simulating the methods of the present invention (FIGS. 1D, 2D, and3). The processor 91 executes the computer code 97. The memory device 94includes input data 96. The input data 96 includes input required by thecomputer code 97. The output device 93 displays output from the computercode 97. Either or both memory devices 94 and 95 (or one or moreadditional memory devices not shown in FIG. 4) may be used as a computerusable medium (or a computer readable medium or a program storagedevice) having a computer readable program code embodied therein and/orhaving other data stored therein, wherein the computer readable programcode comprises the computer code 97. Generally, a computer programproduct (or, alternatively, an article of manufacture) of the computersystem 90 may comprise said computer usable medium (or said programstorage device).

Thus the present invention discloses a process for deploying computinginfrastructure, comprising integrating computer-readable code into thecomputer system 90, wherein the code in combination with the computersystem 90 is capable of performing a method for simulating the methodsof the present invention.

While FIG. 4 shows the computer system 90 as a particular configurationof hardware and software, any configuration of hardware and software, aswould be known to a person of ordinary skill in the art, may be utilizedfor the purposes stated supra in conjunction with the particularcomputer system 90 of FIG. 4. For example, the memory devices 94 and 95may be portions of a single memory device rather than separate memorydevices.

While particular embodiments of the present invention have beendescribed herein for purposes of illustration, many modifications andchanges will become apparent to those skilled in the art. Accordingly,the appended claims are intended to encompass all such modifications andchanges as fall within the true spirit and scope of this invention.

1. A computer program product, comprising a computer usable mediumhaving a computer readable program code embodied therein, said computerreadable program code comprising an algorithm adapted to implement amethod for logic transformation simulation, said method comprising:transforming a gate G into a logic block including an input stage and anoutput stage, wherein inputs of the input stage are the inputs of thegate G, and wherein outputs of the output stage are the outputs of thegate G; and placing each input gate of the input stage in a vicinity ofthe input gate's respective inputs.
 2. The computer program product ofclaim 1, wherein the method further comprises placing each output gateof the output stage in a vicinity of the output gate's respectiveoutputs.
 3. The computer program product of claim 1, wherein saidtransforming the gate G into the logic block comprises: dividing inputsof the gate G into M input groups, at least two proximate inputs;dividing outputs of the gate G into N output groups, each group of the Noutput groups having at least two proximate outputs; and replacing thegate G by (i) M gates in the input stage corresponding to the M inputgroups and (ii) N gates in the output stage corresponding to the Noutput groups, wherein M and N are positive integers, wherein each gateof the M gates is configured to receive input signals from all the theinputs of its corresponding input group, wherein each gate of the Ngates is configured to receive input signals from all the M gates, andwherein each gate of the N gates is configured to send output signals toall outputs of its corresponding output group.
 4. The computer programproduct of claim 1, wherein if the gate G is an OR gate, then all the Mgates and the N gates are OR gates, and if the gate G is an AND gate,then all the M gates and the N gates are AND gates.
 5. The computerprogram product of claim 1, wherein if the gate G is an OR gate, thenall the M gates are NOR gates, and the N gates are NAND gates, and ifthe gate G is an AND gate, then all the M gates are NAND gates, and theN gates are NOR gates.
 6. A method for logic transformation simulation,comprising integrating computer-readable code into a computing system,wherein the code in combination with the computing system is capable of:transforming a two-stage logic into the logic block including an inputstage and an output stage, wherein the two-stage logic includes a firststage and a second stage, wherein the first stage includes T gates, andwherein the second stage includes an output gate that is configured toreceive input signals from all the T gates of the first stage; andplacing each input gate of the input stage in a vicinity of the inputgate's respective inputs.
 7. The method of claim 6, wherein the code incombination with the computing system is further capable of placing eachoutput gate of the output stage in a vicinity of the output gate'srespective outputs.
 8. The method of claim 6, wherein said transformingtwo-stage logic into the logic block comprises: placing the T gates inthe input stage of the logic block; dividing outputs of the output gateinto P output groups, each group of the P output groups having at leasttwo proximate outputs, wherein P is a positive integer; and placing inthe output stage of the logic block P gates corresponding to the Poutput groups, wherein each gate of the P gates is (i) functionallyequivalent to the output gate, (ii) is configured to receive inputsignals from all the T gates of the input stage, and (iii) is configuredto send output signals to all the outputs of its corresponding outputgroup.
 9. The method of claim 8, wherein the code in combination withthe computing system is further capable of, after said placing eachinput gate of the input stage in a vicinity of the input gate'srespective inputs is performed, for each gate Fi, i=1, . . . , T of theT gates of the input stage: dividing inputs of the gate Fi into Qi inputgroups, each group of the Qi input groups having at least two proximateinputs and Ri individual inputs, wherein Qi and Ri are non-negativeintegers but are not both equal to zero; replacing the gate Fi by Qi+1gates, wherein one gate Fi0 of the Qi+1 gates is located where the gateFi was, and wherein the other Qi gates of the Qi+1 gates correspond tothe Qi input groups; and placing each gate of the Qi gates in a vicinityof its corresponding input group, wherein each gate of the Qi gates isconfigured to receive input signals from all the inputs of itscorresponding input group, wherein the gate Fi0 is configured to receiveinput signals from all the Qi gates and from all the Ri individualinputs, and wherein each gate of the P gates of the output stage isconfigured to receive input signals from all the gates Fi0, i=1, . . . ,T.
 10. The method of claim 9, wherein for i=1, . . . , T, if the gate Fiis an OR gate, then all the Qi gates and the gate Fi0 are OR gates, andif the gate Fi is an AND gate, then all the Qi gates and the gate Fi0are AND gates.
 11. The method of claim 9, wherein for i=1, . . . , T, ifthe gate Fi is an OR gate, then all the Qi gates are NOR gates, and thegate Fi0 is a NAND gate, and if the gate Fi is an AND gate, then all theQi gates are NAND gates, and the gate Fi0 is a NOR gate.